Mass flow measurement through rectangular
microchannel from hydrodynamic to near free molecular
regimes
Mustafa Hadj Nacer, Irina Graur, Pierre Perrier
To cite this version:
Mustafa Hadj Nacer, Irina Graur, Pierre Perrier. Mass flow measurement through rectangular
microchannel from hydrodynamic to near free molecular regimes. La Houille Blanche - Revue
internationale de l’eau, EDP Sciences, 2011, pp.49-54. .
HAL Id: hal-01442448
https://hal.archives-ouvertes.fr/hal-01442448
Submitted on 20 Jan 2017
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
La Houille Blanche, n° 4, 2011, p. 49-54
DOI 10.1051/lhb/2011040
DOI 10.1051/lhb/2011040
Mass flow measurement through rectangular microchannel
from hydrodynamic to near free molecular regimes
Mustafa Hadj Nacer1, Irina Graur2, Pierre Perrier3
Université de Provence– Ecole Polytechnique Universitaire de Marseille, Département de Mécanique Energétique, UMR CNRS 6595, 5 rue
Enrico Fermi, 13453 Marseille cedex 13, France
1.
[email protected]
2.
[email protected]
3.
[email protected]
ABSTRACT. — The mass flow rate through microchannels with rectangular cross section is measured for a wide
Knudsen number range (0.0025-26.2) in isothermal steady conditions. The experimental technique called ‘Constant
Volume Method’ is used for the measurements. This method consists of measuring the small pressure variations in the
tanks upstream and downstream of the microchannel. The measurements of the mass flow rate are carried out for three
gases (Helium, Nitrogen and Argon). The microchannel internal surfaces are covered with a thin gold coat of mean
roughness characterized by Ra = 0.87nm (RMS). The continuum approach (Navier-Stokes equations), associated to the
first order velocity slip boundary condition, was used in the slip regime (Knudsen number varies from 0.0025 to 0.1).
The experimental velocity slip and the accommodation coefficients based on the Maxwell kinetic boundary condition
were deduced. In the transitional and near free molecular regimes the linearized kinetic BGK model was used to calculate numerically the mass flow rate. From the comparison of the numerical and measured values of the mass flow rate
the accommodation coefficient was also obtained.
Key-words : gas flows, mass flow rate, pressure measurements, accommodation coefficients.
Mesure de débit massique à travers un microcanal rectangulaire, du régime
hydrodynamique au régime proche moléculaire libre
résumé. — Le débit massique dans un microcanal de section rectangulaire est mesuré pour une large gamme du
nombre de Knudsen (de 0,0025 à 26,2) dans des conditions isothermes stables. La technique expérimentale appelée
« Méthode à volume constant » est utilisée pour les mesures. Cette méthode consiste à mesurer une faible variation de
pression dans les réservoirs en amont et en aval du microcanal. Les mesures du débit massique sont effectuées pour trois
gaz (Hélium, Azote et Argon). Les surfaces intérieures du microcanal sont recouvertes d’une mince couche d’or caractérisée par une rugosité moyenne Ra = 0.87nm (RMS). L’approche continue (équations de Navier-Stokes), associée à la
condition de glissement de vitesse de premier ordre est utilisée dans le régime de glissement (nombre de Knudsen variant
de 0,0025 à 0,1). Les coefficients expérimentaux de vitesse de glissement et d’accommodation basés sur la condition
cinétique aux limites de Maxwell en sont déduits. Dans les régimes de transition et proche moléculaire libre, le modèle
cinétique BGK linéarisé est utilisé pour calculer numériquement le débit massique. A partir de la comparaison entre les
valeurs numériques et mesurées du débit massique, le coefficient d’accommodation est obtenu.
Mots-clés : écoulement de gaz, débit massique, mesure de pression, coefficients d’accommodation
a
G
H
K
Kn
L
M
m
P
Q
Qm
Qs
R
Ra
S
T
V
W
α
δ
ε
λ
μ
σp
τ
υ
Pressure slope [Pa/s]
Dimensionless mass flow rate
Height of the microchannel [m]
Corrective coefficient for the lateral walls influence
Knudsen number
Length of the microchannel [m]
Mass flow rate [kg/s]
Mass of the gas [kg]
Pressure [Pa]
Dimensionless mass flow rate
Mass flow rate [kg/s]
Slip correction term
Specific gas constant [J/kg.K]
Roughness of the microchannel surface [m]
Non-dimensional mass flow rate
Temperature [K]
Volume of the tanks [m3]
Width of the microchannel [m]
Accommodation coefficient
Rarefaction parameter
Specific relative error due to isothermal assumption
Molecular mean free path [m]
Viscosity coefficient [Ns/m2]
Slip coefficient
Experimental time length [s]
Dimensionless local pressure gradient
49
Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/2011040
DOI 10.1051/lhb/2011040
La Houille Blanche, n° 4, 2011, p. 49-54
I.. INTRODUCTION
the volume of the microchannel in order to ensure that the
microflow parameters are independent of time but remain
nevertheless detectable. The variation of the pressure and
temperature during the experiments are measured and the
mass flow rate is deduced from the gas state equation. The
mass variations occurring in the tanks during the experiments do not call into question the stationary assumption.
The experimental setup shown in Fig. 1 takes into account
these constraints : on the volumes of the tanks and on the
pressure and temperature stabilities.
The application of the Micro-Electro-Mechanical-Systems
(MEMS) grows year by year in different fields such as
medical, chemical and other high technological fields. The
flow of the fluid through a channel represents very often an
essential part of such micro-devices. This is the reason why
experimental studies of the microflow properties in channel
geometries appeared in the last ten years [2, 3], [5], [14],
[19]. These studies were limited essentially to the slip flow
regime.
Two main experimental techniques were proposed in the
literature for the mass flow rate measurements in microchannels : the liquid drop method [5, 6], [9], [12], [14], [19]
and the constant volume technique [5, 6, 7], [15]. The both
techniques were realized for the isothermal flow conditions.
The tangential momentum accommodation coefficient which
characterizes the gas/wall interaction was deduced from the
measurements. The values of this accommodation coefficient
found in the literature are in the range (0.81) [6, 7], [15, 16].
These values depend on the surface condition (roughness
and cleanliness) and on the wall material [1].
The aim of this paper is the experimental study of an interaction between different gases and a gold covered silicon
surface. This study is carried out by measuring the mass
flow rate through a rectangular microchannel with a gold
deposit on the internal surface. The constant volume technique is used for the measurements and the microchannel
has an aspect ratio (Height/width) equal to H/w = 0.53.
The working gases are Helium, Argon and Nitrogen.
The Knudsen number range investigated in this work
(0.0025– 26) covers all the flow regimes, from the hydrodynamic to near free molecular regimes. The analytical
and numerical calculations of the mass flow rate using the
Stokes equation in the hydrodynamic and slip regimes are
performed. The simulations based on the numerical solution of the linearized BGK kinetic model in the transitional
and near free molecular regimes are fulfilled. The slip and
accommodation coefficients were deduced by comparing
the experimental data of the mass flow rate to the numerical
results.
The experimental methodology is the following : first, the
experimental loop (Fig. 1) is vacuumed using the VP pump
by opening all the valves of the experimental set up during
a time period (from one to two hours). Then, the valve V4
is closed and the system is filled with the working gas from
the high pressure tank until having the desired inlet pressure
value (pin). After that, the valves V1 and V2 are closed and
the valve V4 is opened again to decrease the pressure in the
outlet tank until obtaining the desired value (pout). Finally,
the valves V3 and V4 are closed and the acquisition of the
temperature and pressure in both tanks can be started. The
duration of the data acquisition depends on the flow rate
about few seconds for high mass flow rate (10-8kg/s) and
250s for low mass flow rate (10-13kg/s).
The details about the connections required in the gas circuit and the estimation of leakage may be found in [6].
The experiments are performed within a narrow temperature range, excluding any heat source in the environment.
During each experiment, the temperature is not maintained,
but controlled to be sufficiently constant to justify the isothermal assumption as quantified in the following section.
It is also to note that the temperature variation due to the
gas expansion in the outlet tank is negligible due to low gas
velocity in the microchannel.
The registration of the pressure pin and pout is carried out
using two pressure transducers C1 and C2 (see Fig. 1) chosen according to their pressure range and connected to the
upstream and downstream tanks, respectively.
The technical characteristics of the pressure transducers
are given in Table 1. The temperature of the gas is assumed
to be equal to the room temperature. Before starting experiments we wait to reach the stabilization of temperature.
II. EXPERIMENTS
The microchannels used in the experiments are fabricated
from silicon with a gold deposit on the internal surfaces. The
height of the microchannel is important to be measured with
good precision because this quantity, set to the power three,
is used in the analytical expression of the mass flow rate.
The dimensions of the microchannel are measured using
an optical microscope and are : Height H = 27.84±0.5μm,
The experimental setup is represented schematically in
Fig. 1. The methodology used to measure the mass flow rate
involves the use of two constant volume tanks connected by
the microchannel and is called “constant volume technique”.
The two volumes have to be much sufficiently greater than
Figure 1 : Schematic of the experimental setup.
50
La Houille Blanche, n° 4, 2011, p. 49-54
DOI 10.1051/lhb/2011040
Table 1 : Technical data for the pressure transducers (CDG capacitance diaphragm gauges).
Detector (A, B, C and D)
Full Scale FS (Pa)
133,322.0 (A)
13,332.2 (B)
1,333.22 (C)
133.32 (D)
0.20 % of reading
0.0050 FS/K
0.01 % of reading/k
0.0015 % FS
Accuracy
Temperature effect on zero
Temperature effect on span
Resolution
To determine the coefficients a and b, we have used a least
squares method. The number of the measured pressure
values varies between 40 and 2000 depending on the tank
pressure. The standard deviation of the coefficient a is calculated following the method used in [6]. It is found to be less
than 0.5%. Therefore, the uncertainty on the measurement of
the mass flow rate, calculated from
width w = 52.23±0.5μm and length L = 15.07±0.01mm. The
roughness of the microchannel is 0.87nm RMS (Roughness
Mean Square). Three gases used in the experiments are
Helium, Nitrogen and Argon.
III. MASS FLOW RATE CALCULATION
Using the constant volume technique, the mass flow rate
can be calculated from the ideal gas state equation
is less than
).
(1)
where V represents the tank volume and R is the specific gas
constant. P, T and m are the pressure, the temperature and
the mass of the gas, respectively. Equation can be transformed into
(2)
(3)
,
IV.1. Hydrodynamic and slip regimes
The mass flow rate through a rectangular microchannel
obtained from the Navier Stokes equations with first order
velocity slip condition reads :
(7)
where
,
is the mean Knudsen number,
based on the mean pressure
. In this
theoretical frame, the coefficient A may be presented in the
form :
(4)
This measurement is affected also by a specific relative error
of 2×10−2 due to the variation of the temperature. If we
consider an isothermal flow between two tanks maintained
at pressure pin and pout, respectively, close to constant values
with a small variation of 1%, so the flow may be considered
as steady flow. To determine the mass flow rate, we use the
registered data of the pressure pi at different instants ti. The
stationary assumption can justify physically the implementation of the polynomial expression of first order in ti
,
The Knudsen number, calculated from the mean pressure
between two tanks, ranges from 0.0025 to 26. This range
covers all regimes : from hydrodynamic flow regime to near
free molecular regime. Many different theoretical and numerical approaches were used to study the flow between two
parallel plates [4], [8], [10], [18] but only a few for rectangular microchannels [17]. A brief presentation of the theoretical approach used for the comparison of the measured
and the theoretical values of the mass flow rate is be given
below.
If
is small compared to 1, then
can be considered as the mass flow rate
through the microsystem. The
constant volume method requires a high-thermal stability.
The deviation of the temperature from the initial value is
smaller than
. The relative variation of the temperature
is in order of
against
for the relative variation of the pressure
,
is clearly less than
. Thus, the mass flow rate
can be written in the
following form
(
IV. BACKGROUND THEORY
As it was shown in [2], [6], the previous relation may be
written as follows
(6)
(8)
where
is the velocity slip coefficient and
is the
corrective coefficient taking into account the influence of
the lateral walls [17] which may be calculated using the
following expression
(5)
51
(9)
DOI 10.1051/lhb/2011040
La Houille Blanche, n° 4, 2011, p. 49-54
Qs is the slip correction term, which depends on the ratio
(
). Furthermore, the non-dimensional mass flow rate
may be deduced from :
with
(10)
which is the mean value of
along the channels does
not depend on the local pressure gradient, but only on its
mean value. The relation and is used to calculate numerically the dimensionless mass flow rate
from the
table of
.
Equation may be rewritten in the more compact form :
(11)
. The mass flow rate calculated using the
where
analytical expression will be compared with the appropriate
experimental measured values.
V. RESULTS AND DISCUSSION
The flows of Argon, Nitrogen and Helium have been studied for different flow regimes : from the hydrodynamic
to near free molecular regime. The mean Knudsen number
ranges from
to
. The pressure ratio
was
maintained constant in the hydrodynamic and slip regimes
around
. In the transitional and near free molecular regimes this ratio was increased until
in order to
increase the mass flow rate. The experimental conditions
are summarized in the Table 2. The large Knudsen number
range
investigated in this study was split in
two parts. We consider the first part of this range below
.
It was shown in [6, 7] that the first order slip flow model is
valid for Knudsen number less than
.
IV.2. Transition and free molecular regime
In the transitional flow regime, the kinetic theory based
on the resolution of the Boltzmann equation must be used
to calculate the mass flow rate through a rectangular microchannel. Here the linearized BGK equation is used to reduce
the important computational efforts required to solve the
Boltzmann equation.
We consider a channel with a width greater than a height
(
) and with length
essentially larger than its width
(
). For this kind of channels the end effect can be
neglected. In this case the local pressure gradient, defined
as follows
(12)
Table 2 : Experimental conditions.
. This assumpis always small for any pressure ratio
tion allows us to linearize the BGK kinetic model equation.
Then, the dimensionless mass flow rate
Quantity
)
Mass flow rate (
Inlet pressure (Pa)
Outlet pressure (Pa)
Average Knudsen number Knm
Pressure ratio
(13)
depends mainly on the rarefaction parameter
(17)
Min
Max
0.91
161 000
53.3
11.1
0.0025
1.54
131 866
88 220
26.2
4.9
(14)
In order to obtain the dimensionless mass flow rate Q the
linearized BGK kinetic model equation subjected Maxwell
specular-diffuse boundary condition is solved using the discrete velocity method. This mass flow rate through the channel cross section Q is calculated from the bulk velocity
V.1. Hydrodynamic and slip regimes
In order to estimate the velocity slip coefficient the measured mass flow rate was fitted with a first order polynomial
form of
(15)
by using the least square method detailed in [14]. The coefficient
is obtained by applying the non-linear square
Marquard-Levenberg algorithm to the measured values of
the mass flow rate normalized according to . The uncertainty
on this coefficient was estimated using the standard error.
From the comparison of the theoretical expressions , and
the experimental mass flow rate , the coefficient A may be
expressed in this form :
The linearized BGK equation is solved for a large range of
(
) and for accommothe rarefaction parameter
dation coefficient
equal to
,
,
and the values
of dimensionless mass flow rate
is so obtained as a
function of the rarefaction parameter.
It is difficult to use directly the expression to calculate the
mass flow rate for the measured values of the pressures in
the tanks because the local pressure and the local rarefaction
parameters are unknown. Using the technique proposed in
[18] we can obtain the mass flow rate based on the pressure
values in both tanks by integrating the local mass flow rate
in the following manner
(18)
(19)
The experimental coefficients
for three gases are
given in the Table 3. The influence of the lateral walls is
taken into account by the coefficient K. The experimental
estimation of the velocity slip coefficient is given in Table 3.
These values are calculated using the first order fitting for
(16)
52
La Houille Blanche, n° 4, 2011, p. 49-54
DOI 10.1051/lhb/2011040
V.2. Transition and free molecular regime
Table 3 : Experimental coefficients Aexp obtained from the
first order polynomial fitting.
Gas
The measured values of the mass flow rate in the transitional and near free molecular regimes are given in Fig. 2 in
non-dimensional form according to equation for three gases
(Helium, Nitrogen and Argon). The results were plotted as
function of the rarefaction parameter
which is calculated
from the mean pressure of the two tanks. The solution of the
linearized BGK equation for two values of the accommodation coefficient (
and
) are obtained numerically
for the experimental microchannel aspect ratio
.
These two curves are plotted also in Fig. 2.
The value of the accommodation coefficient is between
and
for all gases, and probably close to
. We
can note that these values are close to the values found in
slip regime. But the exact values can not be obtained from
simple visual comparison. Other methods have to be developed.
From the Fig. 2 one can see that the minimum value of
the rarefaction parameter reached experimentally is
for
Helium and
for Argon and Nitrogen. This difference
comes from the reason that for the same value of the pressure the rarefaction parameter for Helium is 5 times smaller
than for two other gases due to the difference in the molar
mass : the molar mass of Argon and Nitrogen is about ten
times greater than that of Helium, therefore, for the same
pressure the rarefaction parameter is approximately three
times smaller for Helium.
In Fig. 2 we can also remark some fluctuations of the
experimental points, especially for
and
for the value
of the rarefaction parameter δ less than 1. This is due to
the difficulty to do appropriate measurement of pressure at
very hight level of rarefaction (small values of pressure), the
fluctuations of the pressure become then important :
should increase up to 2% and the uncertainty of the measurements could be increased in the same way.
Molar mass
He
N2
Ar
4.00
28.02
39.95
9.74 ± 0.13
10.18 ± 0.08
10.02 ± 0.13
0.026
0.016
0.025
0.993
0.997
0.994
the Knudsen range
. These values are different
from those obtained theoretically using the BGK and the full
accommodation assumption of the molecule at the wall in
[11] (
) and in [4] (
), as the accommodation is not complete in our experimental conditions. The
errors in Table 3 derive from the experimental error in
, see relations , and do not take into account the systematic
uncertainty coming from the errors on the mass flow rate
measurements (see and the errors due to the channel dimension measurements).
The method used to calculate the accommodation coefficient is proposed in [13] taking into account the Knudsen
layer influence. The authors calculated the values of the slip
coefficient using the BGK kinetic model and the Maxwell
diffuse-specular scattering kernel over the whole range of
values of accommodation coefficient
; a simple « modified » expression associating the slip coefficient and the
accommodation coefficients is proposed
(20)
the slip coefficient for
equals 1.016 [4].
where
The accommodation coefficients calculated from the slip
coefficient using are given in Table 3.
VI. CONCLUSIONS
We compare our results for the slip coefficient with the
experimental values obtained in [16]. The authors of [16]
measured the mass flow rates of various monathomic gases
through a rectangular microchannel with a smooth glass
surface with an average roughness of
, or
of channel height. The surface of the microchannel used in
our experiments is very much smoother and has a roughness
of
(i.e. the relative rughness value is less than
of the channel height). Moreover, the aspect ratio
of our channel (
) is different from those used by
[16] equal to
. The values of the slip coefficient found in [16] are represented in Table 3. Considering
the respective experimental differensies quated above, there
is a good agreement between both results. We can not say
that the values obtained for the various gases are rather close
to each other.
Experimental and numerical investigations on the flow
through rectangular microchannel are presented. The
constant volume technique was used to measure the mass
flow rate through rectangular microchannel which have gold
deposits on the interior surfaces. The continuum approach
(Navier-Stokes equations) with a first order velocity slip
boundary conditions was used in the slip regime to obtain
the experimental velocity slip and accommodation coefficients associated to the Maxwell kinetic boundary condition.
In the transitional and near free molecular regimes the linearized kinetic BGK model was used to calculate numerically
the mass flow rate. This mass flow rate was compared with
the measured values and the accommodation coefficient was
also deduced.
Table 4 : Experimental accommodation and slip coefficients.
Present work
Gas
He
N2
Ar
Porodnov et al [16]
Molar mass
4.00
28.02
39.95
1.305 ± 0.018
1.364 ± 0.012
1.342 ± 0.019
0.868 ± 0.007
0.845 ± 0.004
0.853 ± 0.007
53
1.320 ± 0.01
–
1.294 ± 0.02
0.862 ± 0.006
–
0.872 ± 0.010
DOI 10.1051/lhb/2011040
La Houille Blanche, n° 4, 2011, p. 49-54
Figure 2 : Experimental dimensionless mass flow rate calculated according to compared with the numerical solution fulfilled by
the authors for
. the experimental uncertainties are of the order of 4.1%, see
The results of the slip and accommodation coefficients
show a good agreement with the results found by other
authors. The values of the accommodation coefficients were
found close to each other for various gases. That means
that for low roughness, this value depends more on the surface materials (gold) than on the molecular mass of the gas.
Further experiments have to be performed with larger microchannels to highlight the influence of the lateral walls and to
confirm these first results.
[6]Ewart T., Perrier P., Graur I. A. & Méolans J. G. (2006) —
Mass flow rate measurements in gas micro flows. Exps. Fluids.
41 487-498
[7]Ewart T., Perrier P., Graur I. A. &M´Eolans J. G. (2007)
— Mass flow rate measurements in a microchannel, from
hydrodynamic to near free molecular regimes. J Fluid Mech.
584 337–356
[8]Graur I. A., M´Eolans J. G. & Zeitoun D. E. (2006) —
Analytical and numerical description for isothermal gas flows
in microchannels. Microfluid. Nanofluid. 2 64-77
[9]Harley J., Huang Y., Bau H. & Zemel J. (1995) — Gas
flows in microchannels. J.Fluid Mech. 284 257-274
ACKNOWLEDGEMENTS
[10]K arniadakis G. E. & Beskok A. (2002) — Microflows :
Fundamentals and Simulation. Springer
The research leading to these results has received funding from the European Community’s Seventh Framework
Program (ITN – FP7/2007-2013) under grant agreement
n° 215504.
[11]Kogan M. N. (1969) — Rarefied Gas Dynamics. Plenum.
[12]Lalonde P. (2001) — Etude expérimentale d’écoulements
gazeux dans les microsystèmes à fluides. PhD Thesis Report,
Institut National des Sciences Appliquées de Toulouse, France
Acknowledgment to Femto-ST (Besançon-France) for providing the microchannels.
[13]Loyalka S. K., Petrellis N. & Stvorick S. T. (1975) —
Some numerical results for the BGK model : thermal creep
and viscous slip problems with arbitrary accommodation of the
surface. J. Phys. Fluids. 18 1094
REFERENCES AND CITATIONS
[14]Maurer J., Tabelin P., Joseph P. & Willaime H. (2003) —
Second-order slip laws in microchannels for helium and nitrogen. Phys. Fluids. 15 2613-2621
[1]Agrawal A. And Prabhu S. V. (2008) — Survey on measurement of tangential momentum accommodation coefficient.
J. Vac. Sci. Technom.
[15]Pitakarnnop J. Varoutis S. Valougeorgis D. Geoffroy
S. Baldas L. & Colin S. (2010) — A novel experimental
setup for gas microflows. Microfluid Nanofluid. 8 57–72
[2]Arkilic E., Breuer K. & Schmidt M. (2001) — Mass flow
and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437 29-43
[16]Porodnov B. T., Suetin P. E., Borisov S. F. & Akinshin V. D.
(1974) — Experimental investigation rarefied gas flow in different channels. J. Fluid Mech. 64 417-437
[3]Arkilic E. B., Schmidt M. A. & Breuer K. S. (1997) —
Gaseous slip flow in long microchannels. J. Microelectromech.
syst. 6(2) 167-178
[17]Sharipov F. & Seleznev V. (1998) — Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data. 27(3) 657-709
[4]Cercignani C. & Daneri A. (1963) — Flow of a rarefied gas
between two parallel plates. Phys. Fluids. 6 993-996
[18]Sharipov F. (1999) — Rarefied gas flow through a long rectangular channel. J. Vac. Sci. Technol. 17(5) 3062-3066
[5]Colin S., Lalonde P. & Caen R. (2004) — Validation of a
second-order slip flow model in rectangular microchannels.
Heat Transfer Engng. 25(3) 23-30
[19]Zohar Y., Lee S. Y. K., Lee W. Y., Jiang L. & Tong P. (2002)
— Subsonic gas flow in a straight and uniform microchannel.
J. Fluid Mech. 472 125-151
54